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Three-dimensional and higher-dimensional transformations

All of our discussion in this module has been about two-dimensional transformations — transformations of the plane. But much of our reasoning applies to three dimensions, and even to higher dimensions!

Just as a plane transformation is a function $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, a spatial transformation is a function $F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$. Just as a linear plane transformation is given by a $2 \times 2$ matrix, a linear spatial transformation is given by a $3 \times 3$ matrix.

In three dimensions, we can consider translations by 3-dimensional vectors, rotations about a line, reflections in a plane, projections onto a plane, and dilations from a plane.

Similar methods apply as for the two-dimensional case. Just as there is a standard basis $\{(1,0), (0,1)\}$ of $\mathbb{R}^2$, there is a standard basis $\{(1,0,0), (0,1,0), (0,0,1)\}$ of $\mathbb{R}^3$. A linear transformation $T_M$ takes the three standard basis vectors to the three columns of the matrix $M$. We can visualise $T_M$ as taking the tessellation of $\mathbb{R}^3$ by unit cubes, to a tessellation of $\mathbb{R}^3$ by parallelepipeds (the 3-d version of parallelograms), where each parallelepiped is spanned by the three column vectors of $M$. The determinant of $M$ now gives the volume expansion of $T_M$; its sign again tells us if $T_M$ preserves or reverses orientation.

In essence, everything we have said in this module can be generalised to $\mathbb{R}^3$. And in principle it can be generalised to any dimension $\mathbb{R}^n$ — it's just a little harder to visualise!

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